Abstract: A wide variety of least-squares linear regression procedures used in observational astronomy, particularly investigations of the cosmic distance scale, are presented and discussed. The classes of linear models considered are (1) unweighted regression lines, with bootstrap and jackknife resampling; (2) regression solutions when measurement error, in one or both variables, dominates the scatter; (3) methods to apply a calibration line to new data; (4) truncated regression models, which apply to flux-limited data sets; and (5) censored regression models, which apply when nondetections are present. For the calibration problem we develop two new procedures: a formula for the intercept offset between two parallel data sets, which propagates slope errors from one regression to the other; and a generalization of the Working-Hotelling confidence bands to nonstandard least-squares lines. They can provide improved error analysis for Faber-Jackson, Tully-Fisher, and similar cosmic distance scale relations.

Abstract: Part 1 Introduction: Background and objectives overview. Part 2 Some simple analyses: comparisons at individual times response feature analysis individual curve fitting. Part 3 Univariate analysis of variance: the fundamental model Anova calculation of expected mean-squares expected mean-squares by "synthesis" contrasts, compound symmetry and F-tests relaxing assumptions - univariate, modified univariate or multivariate tests? Part 4 Multivariate analysis: models without special covariance structure hotellings testing for polynomial trends Manova. Part 5 Regression models: special case general case structured covariance case some covariance structures. Part 6 Two-stage linear models: random regression coefficients estimation and testing particular aspects examples. Part 7 Crossover experiments: simple 2x2 designs a Bayesian approach to 2x2 designs more complex crossover designs for two treatments crossover trials with a binary response. Part 8 Categorical data: Markov chain models log-linear models linear model methods for group and time comparisons randomization test approaches some special cases. Part 9 Some further topics: some practical matters antedependence tracking nonlinear growth curves non-normal observations. Part 10 Computer software and examples: repeated measure facilities in BMDP, SPSS and SAS example 1 - BMDP program 2V example 2 - BMDP program 2V example 3 - BMDP program 2V example 4 - SPSS program MANOVA example 5 - SPSS program MANOVA.

Abstract: Small-area estimation has received considerable attention in recent years because of a growing demand for reliable small-area statistics. The direct-survey estimators, based only on the data from a given small area (or small domain), are likely to yield unacceptably large standard errors because of small sample size in the domain. Therefore, alternative estimators that borrow strength from other related small areas have been proposed in the literature to improve the efficiency. These estimators use models, either implicitly or explicitly, that connect the small areas through supplementary (e.g., census and administrative) data. For example, simple synthetic estimators are based on implicit modeling. In this article, three small-area models, of Battese, Harter, and Fuller (1988), Dempster, Rubin, and Tsutakawa (1981), and Fay and Herriot (1979), are investigated. These models are all special cases of a general mixed linear model involving fixed and random effects, and a small-area mean can be expr...

Abstract: The theory of generalized multivariate analysis, based on elliptically contoured distributions, represents a great achievement in the field of multivariate analysis. The text discusses estimation of parameters, testing of hypotheses, and linear models employing the method of stochastic representation, rather than following the classical treatments. It is designed as a textbook for a one-semester course at postgraduate level and as a reference source for lecturers and researchers.

Abstract: A review is given of different ways of estimating the error rate of a prediction rule based on a statistical model. A distinction is drawn between apparent, optimum and actual error rates. Moreover it is shown how cross-validation can be used to obtain an adjusted predictor with smaller error rate. A detailed discussion is given for ordinary least squares, logistic regression and Cox regression in survival analysis. Finally, the splitsample approach is discussed and demonstrated on two data sets.

Abstract: A family of covariance models for longitudinal counts with predictive covariates is presented. These models account for overdispersion, heteroscedasticity, and dependence among repeated observations. The approach is a quasi-likelihood regression similar to the formulation given by Liang and Zeger (1986, Biometrika 73, 13-22). Generalized estimating equations for both the covariate parameters and the variance-covariance parameters are presented. Large-sample properties of the parameter estimates are derived. The proposed methods are illustrated by an analysis of epileptic seizure count data arising from a study of progabide as an adjuvant therapy for partial seizures.

Abstract: It makes sense to attribute a definite percentage of variation in some measure of behavior to variation in heredity only if the effects of heredity and environment are truly additive Additivity is often tested by examining the interaction effect in a two-way analysis of variance (ANOVA) or its equivalent multiple regression model If this effect is not statistically significant at the a = 005 level, it is common practice in certain fields (eg, human behavior genetics) to conclude that the two factors really are additive and then to use linear models, which assume additivity Comparing several simple models of nonadditive, interactive relationships between heredity and environment, however, reveals that ANOVA often fails to detect nonadditivity because it has mueh less power in tests of interaction than in tests of main effects Likewise, the sample sizes needed to detect real interactions are substantially greater than those needed to detect main effects Data transformations that reduce interaction effects also change drastically the properties of the causal model and may conceal theoretically interesting and practically useful relationships Ifthe goal of partitioning variance among mutually exclusive causes and calculating "heritability" coefficients is abandoned, interactive relationships can be examined more seriously and can enhance our understanding of the ways living things develop

Abstract: Convergence and stability properties of the Kalman filter-based parameter estimator are established for linear stochastic time-varying regression models. The main features are: both the variances and sample path averages of the parameter tracking error are shown to be bounded; the regression vector includes both stochastic and deterministic signals, and no assumptions of stationarity or independence are requires; and the unknown parameters are only assumed to have bounded variations in an average sense. >

Abstract: The fundamental theory of shear wave velocity was studied in this paper to select proper parameters for the regression model. Previous regression equations were examined, and the most rational regression model was pointed out. Regression equations were recommended for the clayey, silty, and sandy soils of the Taipei basin. The research showed (1) that N‐value was not the best parameter if the effects of soil type and geologic effect were considered, (2) that the multiple regression model with parameters of N‐value and depth implied a form of shear wave velocity with specific physical meaning, but these two parameters were not independent of each other, and (3) that the intrinsically linear model with the parameter of depth was acceptable when the local soils were first classified according to soil type and geologic effect – then its R‐square value was comparable to the R‐square value of the multiple regression model.

Abstract: A new class of monthly time series urban water demand model is proposed. The model postulates that water use is made up of base use and seasonal use; and the latter consists of three components: a potential use that is dependent on temperature in the absence of rainfall, a water use reduction due to rainfall occurrences, and a random component. The proposed model utilizes three observations that were established in recent daily water use studies: (1) “hysteresis” temperature effect: under the same temperature, water use has different levels and response rates (to a unit change of temperature) in different seasons, (2) “dynamic” rainfall effect: a rainfall causes a temporary reduction in seasonal use that diminishes over time, and (3) “state-dependent” rainfall effect: the higher the seasonal use level prior to the occurrence of a rainfall, the more significant the effect is expected. Monthly rainfall effects are derived through the time aggregating and averaging of a daily response model. The model so obtained is nonlinear in structure. The performance of the model is compared with conventional linear models using monthly data in Austin, Texas, from 1975 to 1984. The proposed nonlinear models outperform the linear models in describing seasonal water use variations in terms of adjusted R2, Akaike information criterion value, and ability to estimate the high summer use in dry and wet years.

Abstract: Influence diagnostics for predictions from a normal linear model examine the effect of deleting a single case on either the point prediction or the predictive density function. Instead of deleting cases, we apply the local influence method of Cook (1986) to assess the effect of small perturbations of continuous data on a specified point prediction from a generalized linear model. Based on local perturbations of the vector of responses, case weights, explanatory variables, or the components of one case, the diagnostics can detect different kinds of influence. Some of the diagnostics are illustrated with an example and compared to standard diagnostic methods.

Abstract: A nonlinear mean- and covariance-structure model for one or more groups is constructed. The model subsumes the usual linear model considered in the literature. It is then shown how to estimate the parameters of the model and the asymptotic covariance matrix of the parameter estimates using pseudo-maximum likelihood (PML) estimation. The resulting estimates are strongly consistent under general regularity conditions, provided only that the model for the first two moments is correctly specified. Nevertheless, because the data are not necessarily drawn from a multivariate normal distribution, the usual likelihood ratio tests for model comparisons in mean- and covariance-structure models do not apply. Wald tests and Lagrange multiplier tests may be used to implement such comparisons. Next, the standard results on ML estimation with missing data are extended to the case of PML estimation with missing data, and the results are applied to the model. The approach to the missing-data problem adopted, whic...

Abstract: This paper considers series estimators of additive interactive regression (AIR) models. AIR models are nonparametric regression models that generalize additive regression models by allowing interactions between different regressor variables. They place more restrictions on the regression function, however, than do fully nonparametric regression models. By doing so, they attempt to circumvent the curse of dimensionality that afflicts the estimation of fully non-parametric regression models.In this paper, we present a finite sample bound and asymptotic rate of convergence results for the mean average squared error of series estimators that show that AIR models do circumvent the curse of dimensionality. A lower bound on the rate of convergence of these estimators is shown to depend on the order of the AIR model and the smoothness of the regression function, but not on the dimension of the regressor vector. Series estimators with fixed and data-dependent truncation parameters are considered.

Abstract: Following Jureckova (1981) we introduce a finite-sample measure of performance of regression estimators based on tail behavior. The least squares estimator is studied in detail, and we find that it achieves good tail performance under strictly Gaussian conditions. However, the tail performance of the least-squares estimator is found to be extremely poor in the case of heavy-tailed error distributions. Further analysis of the least-squares estimator with light-tailed errors indicates the strong influence of the design matrix in determining tail performance. Turning to the tail behavior of various robust estimators of the parameters of the linear model, we focus on tail performance under heavy (algebraic) tailed errors. The 11-estimator is seen to be a leading case: we find a simple characterization of its tail behavior in terms of the design configuration and show that a broad class of M-estimators have the same performance. Perhaps most significantly, it is shown that our finite-sample measure of tail performance is, for heavy tailed error distributions, essentially the same as the finite sample concept of breakdown point introduced by Donoho and Huber (1983). This finding provides an important probabilistic interpretation of the breakdown point and clarifies the role of tail behavior as a quantitative measure of robustness. This link is further explored for high-breakdown regression estimators including Rousseeuw's (1984) leastmedian-of-squares estimator.

Abstract: We review recent developments in the modeling and prediction of nonlinear time series. In some cases apparent randomness in time series may be due to chaotic behavior of a nonlinear but deterministic system. In such cases it is possible to exploit the determinism to make short term forecasts that are much more accurate than one could make from a linear stochastic model. This is done by first reconstructing a state space, and then using nonlinear function approximation methods to create a dynamical model. Nonlinear models are valuable not only as short term forecasters, but also as diagnostic tools for identifying and quantifying low-dimensional chaotic behavior. During the past few years methods for nonlinear modeling have developed rapidly, and have already led to several applications where nonlinear models motivated by chaotic dynamics provide superior predictions to linear models. These applications include prediction of fluid flows, sunspots, mechanical vibrations, ice ages, measles epidemics and human speech. 162 refs., 13 figs.

Abstract: In this paper we propose a solution to the model predictive control problem for the case when the model is given by a nonlinear neural network. The solution follows the algorithm proposed by Peterson et al.[11] where the linear DMC is extended to handle nonlinear systems by updating the linear model with a `disturbance due to nonlinearities' term. Simulation results of a reaction in a CSTR are included. Results show the improvement in control of the proposed algorithm over linear DMC.

Abstract: A general theory of software reliability that proposes that software failure rates are the product of the software average error size, apparent error density, and workload is developed. Models of these factors that are consistent with the assumptions of classical software-reliability models are developed. The linear, geometric and Rayleigh models are special cases of the general theory. Linear reliability models result from assumptions that the average size of remaining errors and workload are constant and that its apparent error density equals its real error density. Geometric reliability models differ from linear models in assuming that the average-error size decreases geometrically as errors are corrected, whereas the Rayleigh model assumes that the average size of remaining errors increases linearly with time. The theory shows that the abstract proportionality constants of classical models are composed of more fundamental and more intuitively meaningful factors, namely, the initial values of average size of remaining errors, real error density, workload, and error content. It is shown how the assumed behavior of the reliability primitives of software (average-error size, error density, and workload) is modeled to accommodate diverse reliability factors. >

Abstract: The cross-validation principle is used to address the task of model selection. Assuming that a set of probabilistic models is given or constructed, the derivation of a selection rule via Bayesian predictive densities is discussed. A selection rule is derived for the set of nested normal linear regression models. Conditioned on the assumption that the true model is in the set of examined models, this rule asymptotically yields consistent selection of the true model. Some simulation results to demonstrate the performance of the selection criterion are included. >

Abstract: Residual plots and diagnostic techniques have become important tools in examining the least squares fit of a linear model. In this article we explore the properties of the residuals from a rank-based fit of the model. We present diagnostic techniques that detect outlying cases and cases that have an influential effect on the rank-based fit. We show that the residuals from this fit can be used to detect curvature not accounted for by the fitted model. Furthermore, our diagnostic techniques inherit the excellent efficiency properties of the rank-based fit over a wide class of error distributions, including asymmetric distributions. We illustrate these techniques with several examples.

Abstract: SUMMARY An approach to building models for paired comparisons experiments based on com- parisons of gamma random variables is considered. The probability that one object is preferred to a second object is taken to be the probability that one gamma random variable with shape parameter r is less than a second independent gamma random variable with the same shape parameter but a different scale parameter. This is motivated by considering a point scoring competition as the basis for each paired comparison. Different values of r provide different paired comparisons models including the Bradley-Terry model, having r equal to one, and the Thurstone-Mosteller model, achieved as r tends to infinity. Applications of the models to data sets are discussed. The most common analysis of paired comparisons experiments associates a parameter with each object in the experiment such that the probability that one object is preferred to a second object is a function of the difference between the associated parameters. This linear model approach is described in detail by David (1988). Some nonparametric techniques are discussed by Kendall & Babington-Smith (1939), Remage & Thompson (1966) and Gokhale, Beaver and Sirotnick (1983). A bibliography of the paired com- parisons literature is given by Davidson & Farquhar (1976). The present paper considers a subset of the linear models which are obtained from a physical model of the comparison procedure. We suppose that the outcome of a paired comparison is determined by comparing the waiting time for r events to occur in each of the two processes being compared. Gamma random variables are used to describe these waiting times. The resulting models include the Bradley-Terry (Bradley & Terry, 1952) and Thurstone-Mosteller (Thurstone, 1927; Mosteller 1951) linear models. The following section provides a motivation for models based on gamma random variables. An independent increments gamma process extends the range of applicable models by permitting noninteger as well as integer values of r. The models which are developed for particular values of r are described in ? 3. Relationships with currently used methodologies are described wherever possible. Section 4 describes inference under the gamma model including estimation procedures and goodness-of-fit measures. The gamma approach is applied to several paired comparisons data sets in ? 5.